This is a traditional Irish song arranged by Clannad. It appears on their debut album Clannad released in 1973.

From Irishpage.com

This song is a conversation between a sea maiden and her daughter, Máire. The maiden lost her seal skin when she came ashore years before and married a mere mortal. She raised a family with this mortal man but she must now leave her family, return to the sea, there to swim on forever now that she has found her seal skin again. Her leaving is the reason that the song is so sorrowful a lament.

Comment from Redwolf

The Mary Cinidh referred to in the song is a Merrow (or Selkie, as they call them in Scotland). There are many stories about the seal people in both Irish and Scottish folklore (in some of which, a person can hold one captive by stealing its skin…in others, the merrow can chose to remain on land as a human but, if she ever touches salt water, she’ll be forced to return to the sea and never again walk on land.

In the back story to An Mhaighdean Mhara, Mary is the latter kind of merrow. Her little daughter Máire fell into the sea and, in rescuing her, Mary is forced to return again to her seal shape and never more return to land to be with her family. In the song, her husband and daughter call out to her in grief and shock as she returns to the sea.

“Éirne” may refer to Loch Erne, or it may refer to the Atlantic Ocean (it is an old word for the Atlantic, so it’s hard to tell which is meant).